HYDRODYNAMICS OF PUMPS

by Christopher Earls Brennen     © Concepts NREC 1994

CHAPTER 1.
INTRODUCTION

 

1.1 SUBJECT

The subject of this monograph is the fluid dynamics of liquid turbomachines, particularly pumps. Rather than attempt a general treatise on turbomachines, we shall focus attention on those special problems and design issues associated with the flow of liquid through a rotating machine. There are two characteristics of a liquid that lead to these special problems, and cause a significantly different set of concerns than would occur in, say, a gas turbine. These are the potential for cavitation and the high density of liquids that enhances the possibility of damaging unsteady flows and forces.

 

1.2 CAVITATION

The word cavitation refers to the formation of vapor bubbles in regions of low pressure within the flow field of a liquid. In some respects, cavitation is similar to boiling, except that the latter is generally considered to occur as a result of an increase of temperature rather than a decrease of pressure. This difference in the direction of the state change in the phase diagram is more significant than might, at first sight, be imagined. It is virtually impossible to cause any rapid uniform change in temperature throughout a finite volume of liquid. Rather, temperature change most often occurs by heat transfer through a solid boundary. Hence, the details of the boiling process generally embrace the detailed interaction of vapor bubbles with a solid surface, and the thermal boundary layer on that surface. On the other hand, a rapid, uniform change in pressure in a liquid is commonplace and, therefore, the details of the cavitation process may differ considerably from those that occur in boiling. Much more detail on the process of cavitation is included in later sections.

It is sufficient at this juncture to observe that cavitation is generally a malevolent process, and that the deleterious consequences can be divided into three categories. First, cavitation can cause damage to the material surfaces close to the area where the bubbles collapse when they are convected into regions of higher pressure. Cavitation damage can be very expensive, and very difficult to eliminate. For most designers of hydraulic machinery, it is the preeminent problem associated with cavitation. Frequently, one begins with the objective of eliminating cavitation completely. However, there are many circumstances in which this proves to be impossible, and the effort must be redirected into minimizing the adverse consequences of the phenomenon.

The second adverse effect of cavitation is that the performance of the pump, or other hydraulic device, may be significantly degraded. In the case of pumps, there is generally a level of inlet pressure at which the performance will decline dramatically, a phenomenon termed cavitation breakdown. This adverse effect has naturally given rise to changes in the design of a pump so as to minimize the degradation of the performance; or, to put it another way, to optimize the performance in the presence of cavitation. One such design modification is the addition of a cavitating inducer upstream of the inlet to a centrifugal or mixed flow pump impeller. Another example is manifest in the blade profiles used for supercavitating propellers. These supercavitating hydrofoil sections have a sharp leading edge, and are shaped like curved wedges with a thick, blunt trailing edge.

The third adverse effect of cavitation is less well known, and is a consequence of the fact that cavitation affects not only the steady state fluid flow, but also the unsteady or dynamic response of the flow. This change in the dynamic performance leads to instabilities in the flow that do not occur in the absence of cavitation. Examples of these instabilities are ``rotating cavitation," which is somewhat similar to the phenomenon of rotating stall in a compressor, and ``auto-oscillation," which is somewhat similar to compressor surge. These instabilities can give rise to oscillating flow rates and pressures that can threaten the structural integrity of the pump or its inlet or discharge ducts. While a complete classification of the various types of unsteady flow arising from cavitation has yet to be constructed, we can, nevertheless, identify a number of specific types of instability, and these are reviewed in later chapters of this monograph.

 

1.3 UNSTEADY FLOWS

While it is true that cavitation introduces a special set of fluid-structure interaction issues, it is also true that there are many such unsteady flow problems which can arise even in the absence of cavitation. One reason these issues may be more critical in a liquid turbomachine is that the large density of a liquid implies much larger fluid dynamic forces. Typically, fluid dynamic forces scale like ρΩ2 D4 where ρ is the fluid density, and Ω and D are the typical frequency of rotation and the typical length, such as the span or chord of the impeller blades or the diameter of the impeller. These forces are applied to blades whose typical thickness is denoted by τ. It follows that the typical structural stresses in the blades are given by ρΩ2 D4/τ2, and, to minimize structural problems, this quantity will have an upper bound which will depend on the material. Clearly this limit will be more stringent when the density of the fluid is larger. In many pumps and liquid turbines it requires thicker blades (larger τ) than would be advisable from a purely hydrodynamic point of view.

This monograph presents a number of different unsteady flow problems that are of concern in the design of hydraulic pumps and turbines. For example, when a rotor blade passes through the wake of a stator blade (or vice versa), it will encounter an unsteady load which is endemic to all turbomachines. Recent investigations of these loads will be reviewed. This rotor-stator interaction problem is an example of a local unsteady flow phenomenon. There also exist global unsteady flow problems, such as the auto-oscillation problem mentioned earlier. Other global unsteady flow problems are caused by the fluid-induced radial loads on an impeller due to flow asymmetries, or the fluid-induced rotordynamic loads that may increase or decrease the critical whirling speeds of the shaft system. These last issues have only recently been addressed from a fundamental research perspective, and a summary of the conclusions is included in this monograph.

 

1.4 TRENDS IN HYDRAULIC TURBOMACHINERY

Though the constraints on a turbomachine design are as varied as the almost innumerable applications, there are a number of ubiquitous trends which allow us to draw some fairly general conclusions. To do so we make use of the affinity laws that are a consequence of dimensional analysis, and relate performance characteristics to the density of the fluid, ρ, the typical rotational speed, Ω, and the typical diameter, D, of the pump. Thus the volume flow rate through the pump, Q, the total head rise across the pump, H, the torque, T, and the power absorbed by the pump, P, will scale according to

Q    α    Ω D3   ......(1.1)
H α Ω2 D2   ......(1.2)
T α ρ D5Ω2   ......(1.3)
P α ρ D5Ω3    ......(1.4)

These simple relations allow basic scaling predictions and initial design estimates. Furthermore, they permit consideration of optimal characteristics, such as the power density which, according to the above, should scale like ρ D2Ω3.

One typical consideration arising out of the affinity laws relates to optimizing the design of a pump for a particular power level, P, and a particular fluid, ρ. This fixes the value of D5Ω3. If one wished to make the pump as small as possible (small D) to reduce weight (as is critical in the rocket engine context) or to reduce cost, this would dictate not only a higher rotational speed, Ω, but also a higher impeller tip speed, Ω D/2. However, as we shall see in the next chapter, the propensity for cavitation increases as a parameter called the cavitation number decreases, and the cavitation number is inversely proportional to the square of the tip speed or Ω2 D2/4. Consequently, the increase in tip speed suggested above could lead to a cavitation problem. Often, therefore, one designs the smallest pump that will still operate without cavitation, and this implies a particular size and speed for the device.

Furthermore, as previously mentioned, the typical fluid-induced stresses in the structure will be given by ρΩ2 D4/τ2, and, if D5Ω3 is fixed and if one maintains the same geometry, D/τ, then the stresses will increase like D-4/3 as the size, D, is decreased. Consequently, fluid/structure interaction problems will increase. To counteract this the blades are often made thicker (D/τ is decreased), but this usually leads to a decrease in the hydraulic performance of the turbomachine. Consequently an optimal design often requires a balanced compromise between hydraulic and structural requirements. Rarely does one encounter a design in which this compromise is optimal.

Of course, the design of a pump, compressor or turbine involves many factors other than the technical issues discussed above. Many compromises and engineering judgments must be made based on constraints such as cost, reliability and the expected life of a machine. This book will not attempt to deal with such complex issues, but will simply focus on the advances in the technical data base associated with cavitation and unsteady flows. For a broader perspective on the design issues, the reader is referred to engineering texts such as those listed at the end of this chapter.

 

1.5 BOOK STRUCTURE

The intention of this monograph is to present an account of both the cavitation issues and the unsteady flow issues, in the hope that this will help in the design of more effective liquid turbomachines. In chapter 2 we review some of the basic principles of the fluid mechanical design of turbomachines for incompressible fluids, and follow that, in chapter 3, with a discussion of the two-dimensional performance analyses based on the flows through cascades of foils. A brief review of three-dimensional effects and secondary flows follows in chapter 4. Then, in chapter 5, we introduce the parameters which govern the phenomenon of cavitation, and describe the different forms which cavitation can take. This is followed by a discussion of the factors which influence the onset or inception of cavitation. Chapter 6 introduces concepts from the analyses of bubble dynamics, and relates those ideas to two of the byproducts of the phenomenon, cavitation damage and noise. The issues associated with the performance of a pump under cavitating conditions are addressed in chapter 7.

The last three chapters deal with unsteady flows and vibration in pumps. Chapter 8 presents a survey of some of the vibration problems in pumps. Chapter 9 provides details of the two basic approaches to the analysis of instabilites and unsteady flow problems in hydraulic systems, namely the methods of solution in the time domain and in the frequency domain. Where possible, it includes a survey of the existing information on the dynamic response of pumps under cavitating and non-cavitating conditions. The final chapter 10 deals with the particular fluid/structure interactions associated with rotordynamic shaft vibrations, and elucidates the fluid-induced rotordynamic forces that can result from the flows through seals and through and around impellers.

REFERENCES

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Last updated 12/1/00.
Christopher E. Brennen